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Theorem or3or 38838
 Description: Decompose disjunction into three cases. (Contributed by RP, 5-Jul-2021.)
Assertion
Ref Expression
or3or ((𝜑𝜓) ↔ ((𝜑𝜓) ∨ (𝜑 ∧ ¬ 𝜓) ∨ (¬ 𝜑𝜓)))

Proof of Theorem or3or
StepHypRef Expression
1 excxor 1616 . . 3 ((𝜑𝜓) ↔ ((𝜑 ∧ ¬ 𝜓) ∨ (¬ 𝜑𝜓)))
21orbi2i 877 . 2 (((𝜑𝜓) ∨ (𝜑𝜓)) ↔ ((𝜑𝜓) ∨ ((𝜑 ∧ ¬ 𝜓) ∨ (¬ 𝜑𝜓))))
3 orc 847 . . . 4 (𝜑 → (𝜑𝜓))
4 exmid 860 . . . . 5 (𝜓 ∨ ¬ 𝜓)
5 pm3.2 446 . . . . . 6 (𝜑 → (𝜓 → (𝜑𝜓)))
6 biimp 205 . . . . . . . . . 10 ((𝜑𝜓) → (𝜑𝜓))
7 iman 388 . . . . . . . . . 10 ((𝜑𝜓) ↔ ¬ (𝜑 ∧ ¬ 𝜓))
86, 7sylib 208 . . . . . . . . 9 ((𝜑𝜓) → ¬ (𝜑 ∧ ¬ 𝜓))
98con2i 136 . . . . . . . 8 ((𝜑 ∧ ¬ 𝜓) → ¬ (𝜑𝜓))
109ex 397 . . . . . . 7 (𝜑 → (¬ 𝜓 → ¬ (𝜑𝜓)))
11 df-xor 1612 . . . . . . . 8 ((𝜑𝜓) ↔ ¬ (𝜑𝜓))
1211bicomi 214 . . . . . . 7 (¬ (𝜑𝜓) ↔ (𝜑𝜓))
1310, 12syl6ib 241 . . . . . 6 (𝜑 → (¬ 𝜓 → (𝜑𝜓)))
145, 13orim12d 945 . . . . 5 (𝜑 → ((𝜓 ∨ ¬ 𝜓) → ((𝜑𝜓) ∨ (𝜑𝜓))))
154, 14mpi 20 . . . 4 (𝜑 → ((𝜑𝜓) ∨ (𝜑𝜓)))
163, 152thd 255 . . 3 (𝜑 → ((𝜑𝜓) ↔ ((𝜑𝜓) ∨ (𝜑𝜓))))
17 bicom 212 . . . . . . 7 ((𝜑𝜓) ↔ (𝜓𝜑))
18 bibif 360 . . . . . . 7 𝜑 → ((𝜓𝜑) ↔ ¬ 𝜓))
1917, 18syl5bb 272 . . . . . 6 𝜑 → ((𝜑𝜓) ↔ ¬ 𝜓))
2019con2bid 343 . . . . 5 𝜑 → (𝜓 ↔ ¬ (𝜑𝜓)))
2120, 12syl6bb 276 . . . 4 𝜑 → (𝜓 ↔ (𝜑𝜓)))
22 biorf 896 . . . 4 𝜑 → (𝜓 ↔ (𝜑𝜓)))
23 simpl 468 . . . . . 6 ((𝜑𝜓) → 𝜑)
2423con3i 151 . . . . 5 𝜑 → ¬ (𝜑𝜓))
25 biorf 896 . . . . 5 (¬ (𝜑𝜓) → ((𝜑𝜓) ↔ ((𝜑𝜓) ∨ (𝜑𝜓))))
2624, 25syl 17 . . . 4 𝜑 → ((𝜑𝜓) ↔ ((𝜑𝜓) ∨ (𝜑𝜓))))
2721, 22, 263bitr3d 298 . . 3 𝜑 → ((𝜑𝜓) ↔ ((𝜑𝜓) ∨ (𝜑𝜓))))
2816, 27pm2.61i 176 . 2 ((𝜑𝜓) ↔ ((𝜑𝜓) ∨ (𝜑𝜓)))
29 3orass 1073 . 2 (((𝜑𝜓) ∨ (𝜑 ∧ ¬ 𝜓) ∨ (¬ 𝜑𝜓)) ↔ ((𝜑𝜓) ∨ ((𝜑 ∧ ¬ 𝜓) ∨ (¬ 𝜑𝜓))))
302, 28, 293bitr4i 292 1 ((𝜑𝜓) ↔ ((𝜑𝜓) ∨ (𝜑 ∧ ¬ 𝜓) ∨ (¬ 𝜑𝜓)))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∧ wa 382   ∨ wo 826   ∨ w3o 1069   ⊻ wxo 1611 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3or 1071  df-xor 1612 This theorem is referenced by:  uneqsn  38840
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